Phase-only filter for generating an arbitrary illumination pattern

ABSTRACT

A phase-only filter for approximating a given optical transfer function for monochromatic incoherent light. The filter is designed by solving an integral equation for a phase function, and imposing, on a transparent plate, an optical path length for the incoherent radiation equal, in radians, to the phase function modulo 2π plus an overall constant.

FIELD AND BACKGROUND OF THE INVENTION

The present invention relates to optical filters and, more particularly,to a phase-only filter that enables the generation of any desiredillumination pattern.

A generalized optical system is shown in FIG. 1. Incoherent radiation ofwavelength λ is provided at an input plane 10, and passes through animaging system of focal length F, represented schematically by a lens12, to an output plane 14. Input plane 10 is a distance U from lens 12.Output plane 14 is a distance V from lens 12. The distances U, V, and Fsatisfy the relationship ##EQU1## FIG. 1 also shows the Cartesiancoordinate system used herein. z is the direction of light propagation.x is perpendicular to z, in the plane of FIG. 1. y is perpendicular toboth x and z, and points out of the plane of FIG. 1 at the reader.

The optical system of FIG. 1 is characterized by a coherent transferfunction (CTF) H(ƒ_(x),ƒ_(y)) which is related to the aperturetransmittance function P(x,y) of lens 12 by the relationship

    H(ƒ.sub.x,ƒ.sub.y)=P(-λVƒ.sub.x,-λVƒ.sub.y)                                         (2)

h(x,y), the impulse response of the CTF, is the inverse Fouriertransform of the CTF: ##EQU2## The optical transfer function (OTF) isthe normalized transfer function of the system for incoherentillumination, defined as: ##EQU3## where the normalizing constant k is:##EQU4## See, for example, J. W. Goodman, Introduction to FourierOptics, McGraw-Hill, San Francisco (1968), pp. 102-130; M. J. Beran andG. B. Parrent, Theory of Partial Coherence, Prentice-Hall Inc.,Englewood Cliffs N.J. (1964); L. Mandel and E. Wolf, "Coherenceproperties of optical fields", Rev. Mod. Phys. vol. 37 p. 231 (1965);and P. S. Considine, "Effects of coherence on imaging system", J. Opt.Soc. Am. vol. 56 p. 1001 (1996). The output obtained for the case ofincoherent illumination is: ##EQU5## where I_(do) and I_(gi) are theoutput and input intensities respectively. Equivalently, the OTF of theoptical system is the ratio of the Fourier transforms of the output andinput intensities: ##EQU6## where "" represents a Fourier Transformoperation.

Using equations (2), (4) and (5), the expression for the OTF may berewritten ##EQU7## Equation (8) yields some of the physical propertiesof the OTF: ##EQU8##

Given a particular input intensity profile I_(gi) and imaging system, adesired output intensity profile I_(do) can be obtained by putting asuitable filter 16 between lens 12 and output plane 14, as shown in FIG.2, thereby forcing the optical system to have the corresponding OTF asexpressed by equation (7). Filter 16 typically functions by attenuatingthe light passing through lens 12. Thus, the energy of the output beamis less than the energy of the input beam. This is undesirable in manyapplications, for example in optical computing, in which many imagingsystems may be cascaded, and the intensity of the ultimate output lightmay be too low to be of practical use. If a non-attenuating, all-phasefilter, that would function as a diffractive optical element, could bedesigned and constructed, the output intensity would be reduced from theinput intensity to a smaller extent than is the case with an attenuationfilter. In fact, the output intensity would be substantially the same asthe input intensity. Devices incorporating such filters would functionwith essentially no attenuation losses in the filters.

There is thus a widely recognized need for, and it would be highlyadvantageous to have, a phase only filter corresponding to a desiredoptical transfer function.

SUMMARY OF THE INVENTION

According to the present invention there is provided a method formaking, for an imaging system of aperture (M(u) at a distance V from anoutput plane, a phase-only filter for radiation of wavelength λ whichapproximates an optical transfer function (ƒ) having a Fourier transformχ(x), the phase-only filter being characterized by a phase function w(u)and an impulse response having an intensity I_(h) (x), the methodcomprising the steps of: (a) solving an integral equation

    cosw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)sinΦ=sinw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)cosΦ

for w(u), wherein ##EQU9## and (b) establishing, at at least onelocation in the filter, an optical path length, in radians, through thefilter, equal to a constant plus the phase function, modulo 2π,evaluated at the location.

According to the present invention there is provided a method fortransforming the intensity profile of a beam of radiation of wavelengthλ, from I_(gi) (x) to I_(do) (x), I_(gi) (x) having a Fourier transform_(gi) (ƒ) and I_(do) (u) having a Fourier transform _(do) (ƒ), themethod comprising the steps of: (a) providing a lens having an apertureM(u) at a distance V from an output plane; (b) solving an integralequation

    cosw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)sinΦ=sinw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)cosΦ

for a phase function w(u), wherein ##EQU10## χ(x) is a Fourier transformof _(do) (ƒ)/_(gi) (ƒ), and I_(h) (x) is an intensity of an impulseresponse of a phase-only filter function exp(iw(u)); and (c) providing afilter having therethrough, at at least one location, an optical pathlength, in radians, equal to a constant plus the phase function, modulo2π, evaluated at the location.

According to the present invention there is provided a filter forproviding an imaging system of aperture M(u), at a distance V from anoutput plane, with an optical transfer function approximately equal to(ƒ), for radiation of wavelength λ, the filter comprising asubstantially transparent plate characterized by a laterally varyingoptical path length therethrough, in radians, equal to a constant plus aphase function w(u) modulo 2π, wherein w(u) is obtained by solving anintegral equation

    cosw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)sinΦ=sinw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)cosΦ

wherein χ(x) is a Fourier transform of (ƒ), I_(h) (x) is an intensity ofan impulse response of a product of M(u) with a phase-only filterfunction exp(iw(u)), and ##EQU11##

The mathematical expression for a phase-only filter is exp(iw(u,v)),where u and v are normalized coordinates of the lens plane of lens 12:

    u=-λVƒ.sub.x v=-λVƒ.sub.y  ( 9)

If the aperture of lens 12 is M(u,v), the aperture transmittancefunction P of equation (2) thus becomes

    P(u,v)=M(u,v)exp(iw(u,v))                                  (10)

The filter is realized by adjusting the optical path length, ofradiation of wavelength λ, in a transparent plate to be w(u,v) radiansmodulo 2π plus a constant number of radians. This can be accomplished byvarying the refractive index of the plate laterally, or by changing thesurface profile of the plate. The surface profile can be changed byadding material to a surface of the plate, for example byphotodeposition, or by removing material from a surface of the plate,for example by etching.

The filter of the present invention is designed for a particularwavelength λ of radiation, but in practice w(u,v) varies slowly with uand v, so that the filter tends to be insensitive to wavelengthvariations in the incident radiation.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is herein described, by way of example only, withreference to the accompanying drawings, wherein:

FIG. 1 (prior art) is a schematic diagram of an optical system;

FIG. 2 (prior art) is a schematic diagram of the optical system of FIG.1 including a filter.

FIG. 3 is a schematic cross section through a phase only filter.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention is of a phase-only optical filter which can beused to shape incoherent radiation with less attenuation than in aphase-and-amplitude filter. Specifically, the present invention can beused to approximate any desired optical transfer function.

The principles and operation of a phase-only filter according to thepresent invention may be better understood with reference to thedrawings and the accompanying description.

The phase function w(u,v) is found by solving an integral equation. Forsimplicity, the integral equation will be derived in the one dimensionalcase (dependence on u only). The extension to two dimensions isstraightforward.

In one dimension, the desired OTF is denoted by (ƒ_(x)). The aperture oflens 12 is M(u). A phase-only filter exp(iw(u)) is attached to the lens.Thus the aperture transmittance function P becomes:

    P(u)=M(u)exp(iw(u))                                        (11)

The Fourier transform of the OTF is: ##EQU12## If constant factors areneglected, the intensity of the impulse response of the phase-onlyfilter, I_(h) (x), is: ##EQU13## Note that in equation (13) thenormalization constant k of equation (5) has been omitted because aconstant factor does not affect the shape of the OTF. The OTF obtainedaccording to the present invention may differ from the desired OTF by aconstant factor. The mean square error between χ(x) and I_(h) (x) isdenoted by ε: ##EQU14## For compactness, ψ(u₁,u₂,x) is defined as:##EQU15## Using equation (14) it is easily seen that ##EQU16##

ε is minimized using the calculus of variations. In this approach, smallvariations δ_(w) (u) are imposed on the phase function of the filter,w(u):

    w(u)→w(u)+δ.sub.w (u)                         (17)

The filter w(u) is sought for which ε is independent of δ_(w) (u). Thisis an extremum of ε (in this case a minimum) because the variation iszero. Thus, the filter obtained is the optimal filter in the sense ofminimum mean square error. The basic assumption of the calculus ofvariations is that the phase variations δ_(w) (u) are small enough thatthe variation terms with higher powers (δ_(w) ²,δ_(w) ³, . . . ) areneglected. Under the assumption of the calculus of variation,

    exp i(δ.sub.w (u.sub.1)-δ.sub.w (u.sub.2))!≈1+i(δ.sub.w (u.sub.1)-δ.sub.w (u.sub.2))(18)

Denoting by I.sup.δ_(h) the variations caused in I_(h) by the variationsof the filter's phase, and applying the approximation of equation (18)to equation (13), yields: ##EQU17## In the most common case, the OTF isreal and even. According to the properties of the Fourier transform, χalso is real and even. (If the OTF is not real and even, the derivationis similar but more complicated.) Using this assumption and equation(19) to evaluate the various terms of equation (16) gives: ##EQU18##Inserting equation (19) into the error expression of equation (15) givesan expression for the error that includes the variation term δ_(w) :

    ε.sup.δ =ε+δ.sub.ε     (22)

δ.sub.ε is the error term related to the variation δ_(w) : ##EQU19##(Note that ε is independent of δ_(w).) Because, according to equation(15),

    ψ(u.sub.1,u.sub.2,x)=-ψ(u.sub.2,u.sub.1,x)         (24)

interchanging the integration variables u₁ and u₂ in equation (21)allows equation (21) to be rewritten as: ##EQU20## Changing the order ofthe integration in equation (25) gives: ##EQU21## For the optimalphase-only filter, the right hand side of equation (24) must beidentically zero for all δ_(w). Thus, the integrand of the doubleintegral in brackets, which is a function of u₂, must be identicallyzero for any value of u₂. The phase w(u₂) that makes that integrandidentically zero is the phase of the optimal phase-only filter. Usingthe trigonometric identity ##EQU22## gives ##EQU23## where ##EQU24##

Equation (28) is equivalent to the following equation: ##EQU25## Thisequation may be solved iteratively for the function w as follows:

1. Assume an arbitrary initial w.

2. Calculate I_(h) using equation (13).

3. Get a new w using equation (29).

4. If the procedure has not converged, return to step 2.

This is the simplest algorithm for solving for w. Other faster and moreaccurate algorithms are known in the art. See, for example, thedescriptions of the Steffenson algorithm, the Newton-Raphson method, andthe Secant algorithm in E. Isaacson and H. B. Keller, Analysis ofNumerical Methods, Wiley, N.Y. (1966).

In the numerical iterative solution of equation (29), additionalconsiderations must be taken into account. The OTF is designed by itsshape. Therefore, the designed OTF may differ from the desired OTF by aconstant factor. If this factor is not taken into account, the iterativesolution does not converge. To achieve convergence, the constant factoris introduced into the minimization of ε in the form of anothervariable, c, multiplying I_(h), and ε is minimized with respect to bothδ_(w) and c. The new form of equation (16) is: ##EQU26## Setting thederivative of the integrand of the right hand side of equation (30) withrespect to c equal to zero gives the value of c that minimizes ε:##EQU27## and equation (29) becomes: ##EQU28##

The generalization of equation (28) to two dimensions isstraightforward. The scalar distance x becomes a distance vector (x,y);the scalar distance u becomes a distance vector (u,v); and the scalarwavenumber ƒ_(x) becomes a wavenumber vector (ƒ_(x),ƒ_(y)). The doubleintegrals in equations (28) and (29) become quadruple integrals. Thefunction Φ in the integrand becomes:

    (xu.sub.2 +yv.sub.2 -xu.sub.1 -yv.sub.1)/λV         (33)

Using vectorial notation, equation (28) and the expression for Φ can begeneralized to a form that includes both the one dimensional case andthe two dimensional case: ##EQU29## In the one dimensional case, theintegration variables x, u₁, and u₂ are the scalar distances x, u₁, andu₂ ; the integrals with respect to x and u₁ are single integrals fromnegative infinity to infinity; and the dot between x and (u₂ -u₁) in theexpression for Φ represents scalar multiplication. In the twodimensional case, the integration variables x, u₁, and u₂ are thedistance vectors (x,y), (u₁,v₁) and (u₂,v₂), respectively; the integralsare double integrals from negative infinity to infinity; and the dotbetween x and (u₂ -u₁) in the expression for Φ represents a vectorialinner product.

Referring now to the drawings, FIG. 3 shows a cross section in thex-direction through a portion of a phase-only filter 20 according to thepresent invention. The top portion of FIG. 3 shows a graph of w(x) as afunction of x. Filter 20 is fabricated in a plate of transparentmaterial having an index of refraction such that the wavelength of theincoherent radiation within the plate is λ' (generally not the same asλ), by making the thickness of the plate equal, in wavelengths, tow(x)/2π at a plurality of sampling points x_(j). FIG. 3 shows filter 20at seven such sampling points, x₁ through x₇. The thickness in thez-direction (the direction of light propogation) are λ'w(x₁)/2π throughλ'w(x₇)/2π, as shown. As noted in the summary, the function w isdetermined up to a multiple of 2π and up to an overall constant for theentire filter 20, so that the thickness of filter 20 at point x_(j) moregenerally can be expressed as λ'w(x_(j))/2π+a_(j) λ'+b, where a_(j) isan integer that may depend on index j and b is an overall constantdistance. The profile of filter 20 between sampling points x_(j) neednot be stepped as shown in FIG. 3, although a stepped profile such asshown is easier to fabricate than a continuous profile. In general, theprofile of filter 20 also varies in the y-direction, perpendicular tothe plane of FIG. 3, in accordance with the variation of w in they-direction. Filter 20 may be given the variable thickness thereof by avariety of techniques well-known in the art, for example photodepositionand etching. In use, filter 20 is simply substituted for filter 16 ofFIG. 2.

Alternatively, filter 20 may be a plate of transparent material oflaterally varying index of refraction, with the index of refraction madeto vary such that at a lateral coordinate (x,y), the optical path lengthof the incoherent radiation in the z-direction is substantially equal tow(x,y) modulo 2π radians, up to an overall constant. This laterallyvarying index of refraction may be imposed on the plate by methodswell-known in the art, for example, ion beam implantation.

While the invention has been described with respect to a limited numberof embodiments, it will be appreciated that many variations,modifications and other applications of the invention may be made.

What is claimed is:
 1. A method for making, for an imaging system ofaperture M(u) at a distance V from an output plane, a phase-only filterfor radiation of wavelength λ which approximates an optical transferfunction (ƒ) having a Fourier transform χ(x), the phase-only filterbeing characterized by a phase function w(u) and an impulse responsehaving an intensity I_(h) (x), the method comprising the steps of:(a)solving an integral equation

    cosw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)sinΦ=sinw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)cosΦ

for w(u), wherein ##EQU30## and (b) establishing, at a plurality oflocations in the filter, an optical path length, in radians, through thefilter, equal to a constant plus the phase function, modulo 2π,evaluated at said location.
 2. The method of claim 1, wherein saidintegral equation is solved iteratively.
 3. The method of claim 1,wherein said establishing of said optical path length is effected byvarying a refractive index of the filter.
 4. The method of claim 1,wherein said establishing of said optical path length is effected byaltering a surface profile of the filter.
 5. The method of claim 4,wherein said alteration is effected by photodeposition.
 6. The method ofclaim 4, wherein said alteration is effected by etching.
 7. A method fortransforming the intensity profile of a beam of radiation of wavelengthλ, from I_(gi) (x) to I_(do) (x), I_(gi) (x) having a Fourier transform_(gi) (ƒ) and I_(do) (u) having a Fourier transform _(do) (ƒ), themethod comprising the steps of:(a) providing a lens having an apertureM(u) at a distance V from an output plane; (b) solving an integralequation

    cosw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)sinΦ=sinw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)cosΦ

for a phase function w(u), wherein ##EQU31## χ(x) is a Fourier transformof _(do) (ƒ)/_(gi) (ƒ), and I_(h) (x) is an intensity of an impulseresponse of a phase-only filter function exp(iw(u)); and (c) providing afilter having therethrough, at a plurality of locations, an optical pathlength, in radians, equal to a constant plus said phase function, modulo2π, evaluated at said location.
 8. The method of claim 7, wherein saidintegral equation is solved iteratively.
 9. The method of claim 7,wherein said optical path length is provided by varying a refractiveindex of said filter.
 10. The method of claim 7, wherein said opticalpath length is provided by altering a surface profile of said filter.11. The method of claim 10, wherein said alteration is effected byphotodeposition.
 12. The method of claim 10, wherein said alteration iseffected by etching.
 13. A filter for providing an imaging system ofaperture M(u), at a distance V from an output plane, with an opticaltransfer function approximately equal to (ƒ), for radiation ofwavelength λ, the filter comprising a substantially transparent platecharacterized by a laterally varying optical path length therethrough,in radians, equal to a constant plus a phase function w(u) modulo 2π,wherein w(u) is obtained by solving an integral equation

    cosw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)sinΦ=sinw(u.sub.2)∫dx(χ(x)-I.sub.h (x))∫du.sub.1 M(u.sub.1)cosΦ

wherein χ(x) is a Fourier transform of (ƒ), I_(h) (x) is an intensity ofan impulse response of a product of M(u) with a phase-only filterfunction exp(iw(u)), and ##EQU32##
 14. The filter of claim 13, whereinsaid integral equation is solved iteratively.
 15. The filter of claim13, wherein said optical path length is provided by varying a refractiveindex of the plate.
 16. The filter of claim 13, wherein said opticalpath length is provided by varying a thickness of the plate.
 17. Thefilter of claim 16, wherein said variation is effected byphotodeposition.
 18. The filter of claim 16, wherein said variation iseffected by etching.